algebraic number field containing the nth roots of unity then where the product is over the finite and infinite primes p of the number field, and where (,)p is the Hilbert symbol of the completion at p. Hilbert's reciprocity law follows from the Artin reciprocity law and the definition of the Hilbert symbol in terms of the Artin symbol. Power residue symbol If K is a number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then the power residue symbol () is related to the Hilbert symbol by The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field by putting ()=() where (b) is the principal ideal generated by b. Hilbert's reciprocity law then implies the following reciprocity law for the residue symbol, for a and b prime to each other and to n: See also Azumaya algebra External links HilbertSymbol at Mathworld References Class field theory Quadratic forms David Hilbert